How Do You Find the Momentum Space Wave Function for an Infinite Square Well?

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Discussion Overview

The discussion centers on finding the momentum space wave function for the nth stationary state in an infinite square well, including the process of graphing probability densities for specific energy levels and calculating the expectation value of p². The conversation touches on theoretical aspects, mathematical reasoning, and foundational questions regarding wave functions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Homework-related

Main Points Raised

  • One participant seeks clarification on how to find a wave function and graph probability densities for the first and second energy levels in an infinite square well.
  • Another participant suggests using the Parseval equality and mentions solving the Hamiltonian in coordinate representation before performing a Fourier transform to obtain the wave function in momentum space.
  • A participant expresses confusion about the mathematical terminology and the origin of wave functions, indicating a lack of familiarity with the underlying concepts.
  • Another participant explains that wave functions are derived as solutions to the Schrödinger wave equation, emphasizing the importance of boundary conditions and suggesting that a Fourier transform can be used to relate position and momentum space wave functions.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the mathematical concepts involved, with some seeking foundational knowledge while others provide technical insights. There is no consensus on the best approach to finding the wave function or the specifics of the calculations involved.

Contextual Notes

Participants highlight the need for clarity on mathematical terminology and foundational concepts, indicating that some assumptions about prior knowledge may not hold for all contributors. The discussion also reflects a dependence on specific boundary conditions and the application of Fourier transforms, which may not be universally understood.

Who May Find This Useful

This discussion may be useful for students and individuals seeking to understand wave functions in quantum mechanics, particularly in the context of infinite square wells and the relationship between position and momentum space representations.

RPI_Quantum
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How exactly does one find a wave function? Specifically, I am asked to find the momentum space wave functoin for the nth stationary state in an infinite square well. Then I am to graph the probability density (phi sqaured) for the first and second energy levels. Lastly, I need to use the momentum space wave function to find the expectation value of p^2.

For finding the expectation value, I assume that I need to integrate using p^2 as an operator between phi and phi*, just as I would if I was using the regular position wavefunction. Is this right?

Anyway, what I really need help with is finding a wave function, and graphing the probability densities.
 
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For the last question,yes u can use each of them...But with caustion.Remember the Parseval equality... :wink:

Solve the HE in the coordinate representation and then transfourier the wave function...

Daniel.
 
Hope I don't sound dumb, but I really have no idea about what you are saying. A lot of the math terminology was never taught to me, so even as I am trying to pick it up, some things escape me.

This seems like a fundmental question, which I have no idea about: where does the wave function come from? In the particular question that I am working on I am coming up with a wave equation for a square well.
 
RPI_Quantum said:
This seems like a fundmental question, which I have no idea about: where does the wave function come from? In the particular question that I am working on I am coming up with a wave equation for a square well.

Wavefunctions are found as solutions to the Schrödinger wave equation, subject to the particular boundary conditions of the system you are looking at. These conditions usually include things such as the wavefunction must be continuous as it crosses the boundary between regions of different potential.

As for finding the wavefunction in momentum space instead of position space, what dextercioby meant was that you can perform a Fourier transform between the two spaces, so if you know one the other isn't too hard to find.

There should be plenty of examples on the web if you just google 'square well potential' or something similar, and perhaps check mathworld for info on Fourier transforms. If your well is infinite, be sure to include that since it's a slightly simpler solution.
 

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